Solving the Inequality ( frac{x - 5}{x^2 x - 5x^2 - 4x - 5} leq 0 ): A Comprehensive Guide

In this article, we will guide you through the process of solving the inequality ( frac{x - 5}{x^2 x - 5x^2 - 4x - 5} leq 0 ). We will break down the steps involved in solving such inequalities and provide a clear, step-by-step explanation to help you understand and apply the process effectively.

Step 1: Factor the Denominator

First, we need to factor the quadratic in the denominator, ( x^2 - 4x - 5 ). Let's do this step by step.

Factor ( x^2 - 4x - 5 ):

We rewrite the equation as follows:

( x^2 - 4x - 5 (x - 5)(x 1) )

Now, we can rewrite the original inequality using the factored form of the denominator:

[ frac{x - 5}{(x - 5)(x 1)} leq 0 ]

Step 2: Simplify the Expression

Notice that ( x - 5 ) appears in both the numerator and the denominator. However, when simplifying, we need to be careful about the domain of the inequality. Therefore, we can rewrite the inequality as:

[ frac{1}{(x 1)} leq 0, text{ for } x eq 5 ]

Step 3: Analyze the Factors

Now, let's analyze the factors of the denominator:

Denominator Analysis

The term ( x 1 ) changes sign at ( x -1 ):

When ( x > -1 ), ( x 1 > 0 ) When ( x -1 ), ( x 1 0 ), the expression is undefined. When ( x

Therefore, the sign of the entire expression ( frac{1}{x 1} ) depends solely on the sign of ( x 1 ).

Step 4: Determine the Intervals

Now we consider the intervals based on ( x 1 ):

When ( x When ( x -1 ): The expression is undefined. When ( x > -1 ), the expression is positive.

By combining these intervals, we can see that the inequality ( frac{1}{x 1} leq 0 ) holds true when:

( x

Step 5: Combine Results

Finally, the solution to the inequality ( frac{x - 5}{x^2 x - 5x^2 - 4x - 5} leq 0 ) is:

( x

Therefore, the solution set is:

( (-infty, -1) )

Note that ( x eq 5 ) because the original expression is undefined at this point.

Final Solution: The solution to the inequality is ( boxed{x